Lazar wrote:I've examined the arguments and determined that the answer is maybe.

I can't open that link thanks to my work internet filter, but ask any decent mathematician and the answer is no. 1 is a unit, it is not prime. To define it otherwise messes up the Fundamental Theorem of Arithmetic, aka the fact that each natural number has a unique prime factorisation.

1 is not prime because it is not useful to define it to be prime. If 1 is prime, all the theorems about prime numbers don't start applying to 1, nor do they stop existing all together: they just become theorems about non-unit primes instead. Writing "non-unit prime" all the time is a waste of ink, so we decided that units aren't allowed to be primes. The argument that excluding 1 is arbitrary misses that all mathematical definitions are fundamentally arbitrary.

The page linked in the OP has other problems, too. The "pro" argument suggests that 1 should be prime because it makes the definition simpler. Well, here's another definition of prime: p is prime if whenever p divides ab, it divides at least one of a and b. Without further qualification, this definition includes 1 (and -1) as prime...but also 0! So if the goal is to state definitions as simply as possible, it depends on which particular definition you prefer, and if you prefer this one, you ought to include 0 too.

A prime number is divisible by two numbers, 1 only by one. Obviously the term prime is just a consolation gift for the numbers that will never reach ones mastery at not being divisible by others and 1 would not want to be grouped with the losers. Also not defining 1 as prime just seems to be more convenient.

A prime number is only divisible by 1 and itself. 1 is divisible by 1 and itself. This is a simple definition. Ergo, 1 is prime.

That's good enough for me.

If the simplicity of a definition can be taken as evidence of its validity, then you have bigger problems than prime numbers. (Of course, if you have multiple definitions to choose from which are all valid, it's nice to use the simplest. But only if no meaning is lost.)

if 1 is prime, there will be situations where you'll have to say "prime numbers except 1".if 1 is not prime, there will be situations where you'll have to say "prime numbers and 1".

since, like everything else in mathematics, prime numbers are defined only because it's a convenient definition to have, the best definition of "prime" is the most convenient one (the one that minimizes such exceptions), which is that 1 is not prime.

To answer this question, I turn to the most official guide of language of which I know, Merriam-Webster Dictionary, which has gotten me through a lot of good times and tough scrabble times.

They define a prime number as "a number that can only be exactly divided by itself and by 1". By this very official declaration by THE official source, I conclude that 1 is considered prime since it can be divided exactly by itself and 1. Math just becomes more convenient in many cases when you assume it isn't prime.

heuristically_alone wrote:↶They define a prime number as "a number that can only be exactly divided by itself and by 1". By this very official declaration by THE official source, I conclude that 1 is considered prime since it can be divided exactly by itself and 1.

But then, the primary examples of using language with precision are logic puzzle books, and that's obviously where we should go for guidance on how to read this definition.

And, as those make clear, a statement that says something about "thing A and thing B" also implies that those two things are distinct. Like, a logic puzzle clue of "The person with the hat and the person who owns a dog are both older than the person who lives at #7"... implies that all three people named are different people. Even though, in some senses, the main statement would still be true if there's just one hat-wearing dog-owning person, that's not what the puzzle implies, and knowing that they're separate people is often necessary information to solve the puzzle.

And so, 1 cannot be a prime number, as "itself and 1" must, logically, be different numbers.

[edit]Also, the definition you quoted is the "simple definition" in M-W... it gives the "full definition" as:

Merriam-Webster wrote:↶any integer other than 0 or ± 1 that is not divisible without remainder by any other integers except ± 1 and ± the integer itself

which... seems to be claiming that negative integers can be prime, which... seems even worse than 1 being prime.

Every prime number has itself as a unique prime factor (ignoring 1s if you add them to the set of primes) - the set of prime factors of p is {p}.But 1 doesn't have that property, its set of prime factors is empty, representing the empty product.

1 lacks properties that other prime numbers have, it's exclusion from the set of primes is more than an arbitrary choice. It would be so special, so odd, that its inclusion as a prime makes no sense.The primes are the multiplicative building blocks of the natural numbers. It's this property that makes them useful and they achieve it without the problematic 1.

Nyktos wrote:↶You need to allow negative primes in order to appropriately generalize the concept of prime to rings which don't have a notion of "positive" and "negative".

But you still need some restriction there, or you have (-2)*(-3) = 2*3, and 6 no longer has a unique prime factorisation. Meanwhile, -1 has no prime factors at all.

I think the cleanest definition of the primes over all the integers is "the primes over the counting numbers, and also -1"... and then add the restriction that the exponent of -1 in a prime factorisation is in Z/2... at least then you still have a unique prime factorisation of all nonzero integers.

The short version: negative numbers complicate number theory, yes, but my professional opinion is that the definition in M-W is dumb, and also a butt.

phlip wrote:But you still need some restriction there, or you have (-2)*(-3) = 2*3, and 6 no longer has a unique prime factorisation. Meanwhile, -1 has no prime factors at all.

Neither 1 nor -1 has a prime factor, because they are both units.

6 already doesn't have a unique factorization, as it is both 2*3 and 3*2. "Unique" really means "unique up to reordering". Allowing negative primes or working in a different ring requires that to be further amended to "unique up to reordering and multiplication by units".

Unlike 1 being prime, whether or not negative numbers qualify is something that legitimately depends on what kind of math you're doing, and in that sense M-W is wrong to say that that's "the" definition. It's convenient in some contexts to only count the positive ones. Considering Z as an example of a more general structure, though, negative primes do (and must) count.

You can't discriminate a number just because it doesn't follow all of the properties the other primes might have. It's practically racist. A platypus is a mammal even though it doesn't have most properties most other mammals have. It is considered a mammal because it fits the basic definition.

You can't discriminate a number just because it doesn't follow all of the properties the other primes might have. It's practically racist. A platypus is a mammal even though it doesn't have most properties most other mammals have. It is considered a mammal because it fits the basic definition.

I think it would be reasonable to define "mammal" to exclude platypodes and echidnae. It happens that biologists defined "mammal" such that it includes those, though (and they probably have their reasons for that). Just like mathematicians have defined "prime" to exclude 1 (and they have reasons for that).

heuristically_alone wrote:You can't discriminate a number just because it doesn't follow all of the properties the other primes might have. It's practically racist. A platypus is a mammal even though it doesn't have most properties most other mammals have. It is considered a mammal because it fits the basic definition.

Biologists work at a number of different levels of specificity depending on the situation. Both platypuses and cats are considered mammals because while there are many differences between the two there are also similarities; a platypus and a cat have an awful lot more in common with each other than either does with a goldfish. But there are different types of mammals: cats are placental while platypuses are monotremes. Some interesting properties of cats generalize to all mammals, others only to placentals. (And of course, some generalize to some other level of specificity.)

So sure, you could consider 1 prime, but then much as in the biological case you'd have to subdivide the primes into two categories: 1, and all the rest. And then all of the actually interesting theorems about primes would become theorems about that second subdivision, and nobody would talk about primes anymore. So what's the point?

heuristically_alone wrote:You can't discriminate a number just because it doesn't follow all of the properties the other primes might have. It's practically racist. A platypus is a mammal even though it doesn't have most properties most other mammals have. It is considered a mammal because it fits the basic definition.

Biologists work at a number of different levels of specificity depending on the situation. Both platypuses and cats are considered mammals because while there are many differences between the two there are also similarities; a platypus and a cat have an awful lot more in common with each other than either does with a goldfish. But there are different types of mammals: cats are placental while platypuses are monotremes. Some interesting properties of cats generalize to all mammals, others only to placentals. (And of course, some generalize to some other level of specificity.)

So sure, you could consider 1 prime, but then much as in the biological case you'd have to subdivide the primes into two categories: 1, and all the rest. And then all of the actually interesting theorems about primes would become theorems about that second subdivision, and nobody would talk about primes anymore. So what's the point?

Just want to point out that 1 wouldn't be alone. 1 and -1 would be in that subcategory. Nobody wants to be alone.

The argument I've heard justify it is that each (whole, non-unity) number is a wall built of bricks, multiplied. A unique (ordering aside) combination of bricks. For any prime, there are no suitable bricks to combine, and it's thus a wall of one brick. There is no "1" brick, because a non-unique multiple of 1s can be used to build the suposedly unique wall of brick(s) relevant to that final number. Away from the Land Of Analogies, discounting 1 as a prime number is vital aspect in certain Number Theory conceptualisations. Or, back again, 1 is the ground upon which you place your bricks (in one pile, in seperate piles, in whatever combination and hierarchy you need). 1, itself is "no bricks", as an empty set, whilst 0 is "no ground" (which thus does not care which bricks you have, they cannot form any wall at all).

But some of that probably looks like self-justification of the hypothesis that 1 is not prime, if you're not actually dealing with maths at a level where you know the necessity of this statw of affairs, to remove some kind of "cosmological constant" awkwardness from all the formulae (at least until 'dark maths' becomes obvious. e.g. if ever it may come to light that the formula fprecisely indicates the nth prime as being f(n) gives either f(1)=1 and f(3)=5 or 2 and 7, respectively, without a +1 or -1 'final adjustment'. Although that'd not be proof of the reism of the standpoint, it'd certainly make some people happy. (After already being ecstatic over finding f in the first place!)

Just define a prime as "A prime is a number that has exactly two unique divisors, itself and 1." By simply adding the word "unique" you exclude 1 as a prime, since it does not possess exactly two unique divisors. If itself and 1 are the same, then they are not unique, and thus there are not two of them. It's a simple definition, just as simple (in my mind) as the version without the word "unique", and it's far, far, far more useful to exclude 1, so... there you go.

Surely it is as ridiculous to consider sqrt(-1) "imaginary" because you can't use it to count pieces of chalk as to consider the number 200 imaginary because by itself it cannot express the location of one point with reference to another. -Isaac Asimov

That's fine except you're basically building a specific definition that excludes 1 from the primes. Which is perfectly fine, but it's no different (in meaning and implication) than writing all prime-related theorems as "for all primes (larger than 1)". It's just shorter. I believe the correct answer to the question "Is 1 prime" is "it doesn't matter".

Fieari wrote:Just define a prime as "A prime is a number that has exactly two unique divisors, itself and 1."

This is how I would phrase it if someone tried to argue that 1 was prime. Seeing as math was made with the assumption that 1 was not prime (citing the FTA) and the existence of prime numbers is really just a thing that humans decided was a thing and holds no intrinsic meaning in the universe (beyond the fact that we look at the universe using our system of math), there's no reason that we can't clarify the definition if someone tries to find a counterexample.

I think the real question is whether the zero ring is a field, and whether 1 belongs with the prime powers.

The characteristic of the field of size p^k is p. If 1 is a prime power, then it is p^0 for every p. So, if the field of one element were to exist, shouldn't it have every characteristic or something? Or would it have a characteristic of 1?

I found my old forum signature to be awkward, so I'm changing it to this until I pick a better one.